English

Blow up dynamics for smooth data equivariant solutions to the energy critical Schrodinger map problem

Analysis of PDEs 2011-02-25 v2 Mathematical Physics math.MP

Abstract

We consider the energy critical Schrodinger map to the 2-sphere for equivariant initial data of homotopy number k=1. We show the existence of a set of smooth initial data arbitrarily close to the ground state harmonic map in the scale invariant norm which generates finite time blow up solutions. We give a sharp description of the corresponding singularity formation which occurs by concentration of a universal bubble of energy. The concentration rate is given by λ(t)=κ(u)Ttlog(Tt)2(1+o(1))\lambda(t)=\kappa(u)\frac{T-t}{|\log (T-t)|^2}(1+o(1)) for some κ(u)>0\kappa(u)>0. The detailed proofs of the results will appear in a companion paper.

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Cite

@article{arxiv.1102.4308,
  title  = {Blow up dynamics for smooth data equivariant solutions to the energy critical Schrodinger map problem},
  author = {Frank Merle and Pierre Raphael and Igor Rodnianski},
  journal= {arXiv preprint arXiv:1102.4308},
  year   = {2011}
}

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R2 v1 2026-06-21T17:29:31.800Z